Explore the intricacies of chaos and ergodicity in quantum and classical systems in this comprehensive lecture by Anatoli Polkovnikov. Delve into the stability of quantum matter in and out of equilibrium at various scales, examining mechanisms that constrain many-body dynamics in Hilbert space. Investigate phenomena such as Many-body localization (MBL), Floquet MBL, Dynamical Freezing, and Hilbert Space Fragmentation. Gain insights into the interdisciplinary nature of this field, drawing from Condensed Matter Physics, Statistical Mechanics, Quantum Field Theory, and Mathematics. Discover the implications for quantum technologies and learn about novel phases of quantum matter. This lecture is part of a program aimed at senior PhD students, postdocs, and faculty members working in related fields, offering a unique opportunity to engage with cutting-edge research in non-equilibrium physics.